Instructions for use

Title Characteristics of magnetic island formation due to resistive interchange instability in helical plasma

Author(s) Ueda, R.; Watanabe, K. Y.; Matsumoto, Y.; Itagaki, M.; Sato, M.; Oikawa, S.

Citation Physics of Plasmas, 21(5), 052502-1-052502-8https://doi.org/10.1063/1.4875340

Issue Date 2014-05

Doc URL http://hdl.handle.net/2115/57111

RightsCopyright 2014 American Institute of Physics. This article may be downloaded for personal use only. Any other userequires prior permission of the author and the American Institute of Physics. The following article appeared in Physicsof Plasmas vol.21 and may be found at http://dx.doi.org/10.1063/1.4875340

Type article

File Information 1.4875340.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

Characteristics of magnetic island formation due to resistive interchange instability inhelical plasmaR. Ueda, K. Y. Watanabe, Y. Matsumoto, M. Itagaki, M. Sato, and S. Oikawa Citation: Physics of Plasmas (1994-present) 21, 052502 (2014); doi: 10.1063/1.4875340 View online: http://dx.doi.org/10.1063/1.4875340 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/21/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Three dimensional instabilities of an electron scale current sheet in collisionless magnetic reconnection Phys. Plasmas 21, 062116 (2014); 10.1063/1.4885636 Nonlinear stability of magnetic islands in a rotating helical plasma Phys. Plasmas 19, 122510 (2012); 10.1063/1.4773041 Hyper-resistivity and electron thermal conductivity due to destroyed magnetic surfaces in axisymmetric plasmaequilibria Phys. Plasmas 19, 062502 (2012); 10.1063/1.4728080 Numerical simulations of separatrix instabilities in collisionless magnetic reconnection Phys. Plasmas 19, 042110 (2012); 10.1063/1.3698621 Measurements of the momentum and current transport from tearing instability in the Madison Symmetric Torusreversed-field pincha) Phys. Plasmas 16, 055903 (2009); 10.1063/1.3090325

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

Characteristics of magnetic island formation due to resistive interchangeinstability in helical plasma

R. Ueda,1 K. Y. Watanabe,2 Y. Matsumoto,1 M. Itagaki,1 M. Sato,2 and S. Oikawa1

1Graduate School of Hokkaido University, Sapporo 060-8628, Japan2National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan

(Received 22 January 2014; accepted 15 March 2014; published online 8 May 2014)

Focusing attention on the magnetic island formation, we investigate the characteristics of the

resistive interchange magnetohydrodynamics instabilities, which would limit a high beta operational

regime in helical type fusion reactors. An introduction of a new index, i.e., the ratio of the magnetic

fluctuation level to the radial displacement, enables us to make a systematic analysis on the

magnetic island formation in the large helical device-like plasmas during the linear growth phase;

(i) the interchange instability with the second largest growth rate makes the magnetic island larger

than that with the largest growth rate when the amplitude of the radial displacement in both cases is

almost the same as each other; (ii) applied to a typical tearing instability, the index is smaller than

that for the interchange instability with the second largest growth rate. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4875340]

I. INTRODUCTION

The Large Helical Device (LHD) is the largest helical-

type plasma experimental device, which has achieved a

plasma with a high beta (b) value that is equivalent to a fusion

reactor. The b value, the ratio of the plasma pressure to the

magnetic pressure, is an indicator to evaluate the economic ef-

ficiency. Since helical-type devices do not require a toroidal

current intrinsically, they have an advantage that the current-

driven instability does not appear, and are suitable for steady

state operation. However, the pressure-driven instability such

as the interchange instability appears in helical-type devices.

For high b plasmas in the LHD, since the magnetic hill exists

in the peripheral region, there is always a possibility that the

resistive interchange instability appears. It is known that the

resistive interchange instability causes a degradation of

plasma confinement properties.1 Therefore, it is one of the

crucial issues for the realization of the fusion reactor to study

the characteristics of the resistive interchange instability.

In the LHD experiments, magnetohydrodynamics

(MHD) activities have been observed by use of magnetic

probes, electron cyclotron emission (ECE), and soft X-ray

(SX) measurements.2–4 With these measurements, we can

investigate structures of instabilities, e.g., the toroidal and the

poloidal structure, the radial profile of the radial displace-

ments and the perturbed temperature. According to the meas-

ured magnetic fluctuation resonated with the low-ordered

rational surface, the magnetic fluctuation level increases with

the decrease in the magnetic Reynolds number (the S num-

ber) and the increase in the b value.2 The coherent magnetic

fluctuation observed outside the plasma implies an existence

of the perturbed current resonated with the rational surface. It

is considered that the perturbed currents due to MHD instabil-

ities induce the magnetic island formation. Moreover, due to

the presence of the fluctuations in the marginally unstable

plasma, an apparent degradation in the electron temperature

gradients and in the confinement performance is observed.1

On the other hand, the characteristics of the resistive

interchange instability have been studied from the viewpoint

of the dependence of the linear growth rate and the width of

the radial perturbed profile on the b and the S number, and

from the aspect of their nonlinear saturation behavior as

well. An early theoretical work on the resistive interchange

instability was made by Furth et al.5 They clarified that the

linear growth rate and the width of the radial profile are pro-

portional to S�1=3.5 Other numerical studies also clarified the

characteristics of the resistive interchange instability in the

linear growth phase. For example, Ichiguchi et al.6 carried

out the linear analysis for the heliotron plasma. They showed

that the linear growth rate is proportional to S�1=3 in the low

b plasma where the ideal interchange instability is negligi-

ble. Furthermore, it was shown that the dependence of the

growth rate on the S number deviates from S�1=3 in high bplasma where the ideal interchange instability is definite.6

There are also many simulation studies about the effect of

the resistive interchange instability in the nonlinear satura-

tion phase on the plasma confinement property.7–11 The char-

acteristics of the perturbed current due to the resistive

interchange instability and the consequent magnetic island

have been reported, although the reports are for the limited

range of the b value and the S number. However, it is

increasing for the demand of the systematical prediction of

the characteristics of the resistive interchange instability,

e.g., the magnetic island formation property, to interpret the

LHD experiments results.12,13 The reasons for this are as fol-

lows. The ranges of the b and the S numbers covered in the

LHD experiments have become wider, and the measurement

accuracy of the perturbation profile has been improved.

These improvements enable us to measure systematically the

plasma pressure profile with crucial effect on the instability

and the rotational transform profile, and enable us to expand

the experimental database of the appearance domain for the

resistive interchange instability.

1070-664X/2014/21(5)/052502/8/$30.00 VC 2014 AIP Publishing LLC21, 052502-1

PHYSICS OF PLASMAS 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

In this study, we propose a new index which is made up

of a combination of the magnetic fluctuation level and the

amplitude of the radial displacement. The index enables us

to investigate the behavior of the magnetic island formation

even in the linear growth phase. This investigation has been

systematically performed against a configuration to model

the LHD plasma in the linear growth phase and also in a

wide range of the b value and the S number. We focus on the

m/n¼ 1/1 mode instability resonated with the peripheral

rational surface, because such low-ordered mode is observed

in a wide range of b value in the LHD experiments. Here, mand n are the poloidal and the toroidal mode numbers,

respectively. Our analysis is carried out based on an eigen-

value problem approach. We investigate not only the charac-

teristics of the first eigenfunction but also the characteristics

of the second and the third eigenfunction. The first, the sec-

ond, and the third eigenfunction correspond to the largest,

the second largest, and the third largest growth rates,

respectively.

II. NUMERICAL MODELS

A. Model equations

The following linearized equations are employed in this

study:

@~w@s¼ rk~/ þ �gr2

?~w; (1)

@

@sr2?

~/ ¼ rkr2?

~w þrJ0 �r~w � ef þrX�r~p � ef; (2)

and

@~p

@s¼ r~/ �rp0 � ef: (3)

Equations (1)–(3) are based on the reduced MHD equa-

tions14 described in cylindrical coordinates (q, h, f). The

variables ~w, ~/, and ~p are the perturbed components of the

poloidal flux, the stream function, the plasma pressure

normalized by �aB0, �aVA, and �B20l0, respectively. �g � g=

ðl0�aVAÞ denotes the normalized resistivity. Time and

radius are also normalized as s � t=ða=ð�VAÞÞ and q� r/a,

respectively. � is the inversed plasma aspect ratio

(� a/R0). Here, a and R0 are the plasma minor radius and

the major radius, respectively. J0 and p0 are the normalized

equilibrium toroidal current and the normalized equilib-

rium pressure. Here, VA is the Alfv�en velocity. The mag-

netic Reynolds number (S number) is defined as S ¼ �g�1.

The operators rk and r? are defined as rk � ðB0=B0Þ � rand r2

? � r2 � @2=@f2. The contribution of the magnetic

curvature is expressed using X in Eq. (2), which is given

by

X0 ¼ N�

‘

1

q2ðq4iÞ0; (4)

with 0 � d/dq. N is the pitch number and ‘ is the pole num-

ber. In the LHD, N¼ 10, ‘¼ 2, and �¼ 0.6/3.6. The symbol i

denotes the rotational transform. With a Fourier-Laplace

transformation, the perturbed components ð~w; ~/; ~pÞ are

expressed as ðw; /; pÞexpðiðmh� nfÞ � ixsÞ. We assume

that both ~w and ~p have only real part, while both ~/ and xhave only imaginary part. Substituting these expressions to

Eqs. (1)–(3), we obtain

cw ¼ �kk/ þ �gr2

?w; (5)

cr2

?/ ¼ kkr2

?w þm

qðX0p þ J00wÞ; (6)

and

cp ¼ m

qp00/; (7)

where

kk ¼ mi� n; (8)

r2

? �1

qd

dqq

d

dq� m2

q2: (9)

It is noted that x has been replaced by ic, where c is the lin-

ear growth rate normalized by the poloidal Alfv�en time.

Equations (5)–(7) are solved as an eigenvalue problem with

c being the eigenvalue.

B. Numerical methods

Assuming the LHD plasmas to be current-less in our

interchange instability analysis, we ignore the equilibrium

current J0. The boundary conditions

w ¼ 0 and / ¼ 0 (10)

are set on the position q¼ 0 and q¼ 1. Additionally, we give

the boundary condition w0ðq ¼ 1Þ ¼ 1 in order to remove a

trivial solution. Under the boundary conditions except for

wðq ¼ 1Þ ¼ 0, we solve Eqs. (5)–(7) with given initial

approximation c, and iteratively update c until the boundary

conditions on q¼ 1 are satisfied. The eigenfunction of p is

determined by Eq. (7). The derivative with respect to q is

obtained using the second-order accurate central-difference

scheme. The interval 0�q� 1 is divided equidistantly into

5000 grids. We investigate the characteristics of the instabil-

ities from the radial perturbation profiles of the poloidal flux,

the current density, and the radial displacement. Using the

radial perturbation profiles of w; /, and p, we can express

the radial profile of the perturbed current density and the

radial displacement as

j ¼ r2

?w (11)

and

nr ¼ �m/cq

; (12)

respectively.

052502-2 Ueda et al. Phys. Plasmas 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

C. Equilibrium profile

Typical equilibrium profiles

p0 ¼1

2�2ð1� paÞð1� q2Þ2 þ pa

h ibc; (13)

and

i ¼ � 1

qdw0

dq¼ ic þ ðiw � icÞq2; (14)

are used for the interchange instability analysis. We assume

pa¼ 10�3, ic ¼ 0:461, and iw ¼ 1:561. The symbol bc here

denotes the central beta, the value of which is changed vari-

ously for parameter survey. Figure 1 shows the rotational

transform and the pressure profile for bc¼ 4%. The

(m, n)¼ (1, 1) rational surface is located at q¼ 0.7.

III. CHARACTERISTICS OF RESISTIVE INTERCHANGEINSTABILITY

A. Radial structure

In the plasmas with a finite resistivity, the reconnection

of the magnetic lines can occur, so that the magnetic islands

can be formed. Focusing attention on the magnetic island

formation, we investigate the characteristics of the resistive

interchange instability. Figure 2 shows the radial profile of

(a) the perturbed poloidal flux, (b) the perturbed current den-

sity, and (c) the radial displacement for the first eigenfunc-

tion of the resistive interchange instability with S¼ 104,

bc¼ 2%. The vertical solid line denotes the location of the

resonated rational surface i ¼ 1. The normalized growth rate

is c¼ 2.64� 10�2. The Suydam criterion, the stable limit of

the ideal interchange instability, is given by

1

4

didq

� �2

þ 1

q2

dp0

dqdXdq

< 0: (15)

Using this criterion, we obtain the stability limit of the ideal

interchange instability as bc¼ 2.3%. Since the calculation

condition bc¼ 2% of Fig. 2 is below the stability limit, the

radial profiles represent the typical structures of the resistive

interchange instability. It can be seen from Fig. 2(a) that

although the perturbed poloidal flux profile has almost the

odd structure with respect to the rational surface, the w value

on the rational surface is not exactly zero. The magnetic field

is expressed as

B ¼ ef �rwþ B0ef: (16)

Due to the non-zero w, we know Br 6¼ 0 on the resonant

rational surface, which leads to the magnetic island forma-

tion. We can see from Fig. 2(b) that the perturbed current in

radial profile has the opposite directions with respect to the

rational surface. The radial displacement shown in Fig. 2(c)

has almost the even structure, which means the displace-

ments are the same direction in the vicinity of the rational

surface. Next, we describe the characteristics of the eigen-

function with growth rate smaller than the first eigenfunc-

tion, although they have the same poloidal and toroidal

numbers as the first eigenfunction. For the resistive inter-

change instability, the growth rates of the second and the

third eigenfunctions are the same order as the first eigenfunc-

tion, which will be shown in Sec. III B. Figures 3 and 4 show

the radial profile of the second and the third eigenfunctions,

respectively. Figures 3 and 4 as well as Fig. 2 show (a) the

perturbed poloidal flux, (b) the perturbed current density,

and (c) the radial displacement. These results are calculated

for the same plasma that is shown in Fig. 2. The growth rate

of the second and the third eigenfunctions are

c¼ 1.67� 10�2 and c¼ 8.09� 10�3, respectively. It can be

seen from Fig. 3(a) that the radial profile of the perturbed

poloidal flux is almost the even structure with respect to the

rational surface. The w value on the rational surface is finite,

which is quite different from that of the first eigenfunction.

The radial perturbed current density profile of the second

eigenfunction shown in Fig. 3(b) has almost the even struc-

ture with respect to the rational surface, which is considered

to form the magnetic island easily. The directions of the cur-

rents inside and outside the rational surface are opposite to

that on the surface. The radial displacement profile of the

second eigenfunction shown in Fig. 3(c) has almost the odd

structure with respect to the rational surface. In the third

eigenfunction, the poloidal flux profile shown in Fig. 4(a)

represents almost the odd structure with respect to the

FIG. 1. Equilibrium profile of the pressure (solid line) and the rotational

transform (dashed line). Vertical line denotes the position of i ¼ 1 rational

surface.

FIG. 2. Radial profile of the eigenfunc-

tions (a) w, (b) j, and (c) nr with the

largest growth rate. Vertical solid line

denotes the position of i ¼ 1 rational

surface.

052502-3 Ueda et al. Phys. Plasmas 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

rational surface. Although this structure looks similar to that

of the first eigenfunction, the radial profile of the perturbed

current density shown in Fig. 4(b) represents an additional

oscillation and different structure from that of the first eigen-

function. The radial displacement profile shown in Fig. 4(c)

represents approximately the even structure. The directions

of the radial displacements inside and outside the rational

surface are opposite to that on the surface. From the radial

displacement profiles shown in Figs. 2–4(c), we can find that

the first, the second, and the third eigenfunction have one,

two, and three extremal points, respectively.

B. Dependence of growth rate and width of radialstructure on b and magnetic Reynolds number

It is well known that the plasma pressure and the S num-

ber affect the linear growth rate and the width of the radial

perturbation structure.1 In this section, we investigate these

effects for the first, the second, and the third eigenfunctions.

Figure 5 shows the linear growth rates of each eigenfunction

as a function of the S number for various b plasmas with

2%<bc< 5%. For low b value and small S number, the lin-

ear growth rates of the second and the third eigenfunctions

are the same order as those of the first eigenfunction. For

high b and large S number, however, the growth rates of the

first eigenfunction are much larger than those of the second

and the third eigenfunctions. It is well known that the growth

rate depends on the S number as c / S�1=3.5 We can see

from Fig. 5 that the dependence of c on S is close to c/ S�1=3 in low b. The above dependence implies that not

only the first eigenfunction but also the second and the third

eigenfunctions have the characteristics of the resistive inter-

change instability. For the bc> 3% plasmas, the dependence

deviates from c / S�1=3. For high b regime, in particular, the

S-dependence disappears. This is because the ideal inter-

change instability appears over the Suydam limit, i.e.,

bc¼ 2.3% for the present equilibrium profile, and becomes

gradually dominant with the increase in the b value. The

similar conclusion is already shown by Ichiguchi et al.6 On

the contrary, the second and the third eigenfunctions keep

the characteristics of the resistive interchange instability, i.e.,

c / S�1=3 even in high b regime. The reason for the clearly

appearance of such characteristics is that the growth rates of

the second and third eigenfunctions in the ideal interchange

instability are much smaller than that of the first eigenfunc-

tion as shown in the Appendix. Next, we show in Fig. 6 that

the dependence of the width of the radial displacement

profile on the S number in the first, the second, and third

eigenfunctions. Here, the radial width is evaluated from the

FWHM (full width at the half maximum) of the radial dis-

placement profile. Since the second and the third eigenfunc-

tions have multiple extremal points, the FWHM is evaluated

for the maximum value of the jnr j. It can be seen from Fig. 6

that the first, the second, and the third eigenfunctions repre-

sent almost the same S-dependence of the FWHM, i.e.,

FWHM / S�1=3 in low b values. The S-dependence deviates

from FWHM / S�1=3 in high b regime. This is because the

ideal interchange instability becomes dominant as well as

the S-dependence of the growth rate. For the second and the

third eigenfunctions, the deviation from the S-dependence of

the FWHM is not seen even in large S number. Also, the

S-dependence differences of the FWHM among the first, the

second, and the third eigenfunctions are not significantly rec-

ognized in low b regime and also in a wide range of the Snumber. However, we should note that because the second

eigenfunction has the two peak values, there is a possibility

that the sum of the FWHMs for each peak is larger than the

FWHM of the first eigenfunction. Then, the radial region

influenced by the second eigenfunction can be wider than

that by the first eigenfunction. Similar notices can also be

given for the third eigenfunction.

C. Introduction of an index on the magnetic islandformation

To investigate the characteristics of the magnetic island

formation in the linear growth phase of the interchange insta-

bility, we focus on the ratio of the magnetic island width to

the radial displacement. In this section, the first and the sec-

ond eigenfunctions are analyzed as examples of an odd and

an even structure. Using the normalized poloidal flux and the

shear parameter i0 on the rational surface, we can express the

magnetic island width normalized by the minor radius as

FIG. 3. Radial profile of the eigenfunc-

tions (a) w, (b) j, and (c) nr with the

second largest growth rate. Vertical

solid line denotes the position of i ¼ 1

rational surface.

FIG. 4. Radial profile of the eigenfunc-

tions (a) w, (b) j, and (c) nr with the

third largest growth rate. Vertical solid

line denotes the position of i ¼ 1

rational surface.

052502-4 Ueda et al. Phys. Plasmas 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

�w ¼ 4

ffiffiffiffiffiffiffijwji0

s ����s

: (17)

Here, subscript s denotes the rational surface. We cannot

directly evaluate the magnetic island width by Eq. (17), since

the amplitude of w on the rational surface cannot be obtained

in the linear analysis. On the other hand, we can see ws / �w2

from Eq. (17) and / / nr from Eq. (12), respectively. Since

the ratio of w to / is kept in the linear growth phase, the ra-

tio �w2=nr could be a good index to characterize the magnetic

island formation during the linear growth phase. In the index,

we take the maximum values of jnrj as the reference nr,

which is near the rational surface as shown in Sec. III A.

Finally, a new index to characterize the magnetic island for-

mation is defined as

In �w2

nr;peaka¼ 16c

m

jwji0

� �s

q

/

� �peak

: (18)

Figure 7 shows the variations of In as a function of the Snumber for the first and the second eigenfunctions. In for the

first eigenfunction of the interchange instability (¼ In,I1) is

approximately proportional to S�1=3 in low b. Meanwhile, in

high b and large S number, In,I1 is approximately propor-

tional to S�1. In for the second eigenfunction of the inter-

change instability (¼In,I2) is proportional to S�1=3, which can

be seen in a beta range wider than In,I1. This is because the

linear growth rate of the second eigenfunction for the ideal

interchange instability is much smaller than that of the first

eigenfunction. Also, we can see that In,I2 is larger than In,I1,

which shows that the second eigenfunction forms the mag-

netic islands larger than the first eigenfunction if the ampli-

tudes of the jnr;peakj in both cases are comparable.

FIG. 5. Linear growth rates of the eigenfunctions with (a) the largest, (b) the

second largest, and (c) the third largest growth rates for various b plasmas as

function of S. Solid line denotes c / S�1=3.

FIG. 6. Linear FWHMs of nr with (a) the largest, (b) the second largest, and

(c) the third largest growth rates for various b plasmas as a function of S.

Solid line denotes FWHM / S�1=3.

052502-5 Ueda et al. Phys. Plasmas 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

IV. DISCUSSION

In Sec. III, we studied the aspect of the magnetic island

formation due to the resistive interchange instability, and

proposed the new index In in Eq. (18). In this section, using

the index In, we compare the characteristics between the

resistive interchange instability and the tearing instability.

We also study the characteristics of these instabilities by

using another characteristic index on the magnetic island for-

mation. The index is defined as Iw � ws=wpeak (/ w2), which

can be obtained from only the perturbed poloidal flux. Here,

wpeak denotes the maximum absolute value of the perturbed

poloidal flux.

To analyze the characteristics of the magnetic island

formed by the tearing instability, the magnetic curvature

term mq X0p in Eq. (6) is ignored. Instead, we include the equi-

librium current density term mq J00w that was once ignored for

the interchange instability analysis. Figure 8 shows the radial

profiles of (a) the perturbed poloidal flux, (b) the perturbed

current density, and (c) the radial displacement of the typical

tearing instability. Here, we applied the calculation condi-

tions, J0 / ð1� q2Þ�; S ¼ 104; � ¼ 3; ic ¼ 0:9; iw ¼ 0:225,

and m/n¼ 2/1. We can see from Fig. 8(a) that the perturbed

poloidal flux of the tearing instability has the clearly finite

value on the resonant rational surface. The perturbed current

density profile shown in Fig. 8(b) represents the negative

maximum value outside the rational surface. On the contrary,

the positive current flows inside the rational surface. From

the radial displacement profile shown in Fig. 8(c), we

observe two peaks which have opposite signs with each

other. The peak inside the rational surface is much larger

than that outside the surface.

Using the index In proposed in the Sec. III C, we make a

comparison between the characteristics on the magnetic

island formation due to the resistive interchange instability

and those due to the tearing instability. Figure 9 shows In as

a function of the S number. It can be seen that In,I2 is the

largest index in a wide range of the b value and the S num-

ber. The index In for the tearing instability (¼ In,T) is clearly

smaller than In,I2. In small S number, In,T is comparable to

In,I1. For high b and large S number plasmas with the domi-

nant ideal interchange instability, In,I1 is much small index.

These results show that the magnetic island width of the

resistive interchange instability is not always smaller than

that of the tearing instability if the radial displacements in

both cases are comparable. In the LHD experiments, we

observed 5%–40% radial displacement normalized by the

minor radius for the interchange instability.1,13 Also, the

MHD activities are observed around 105< S< 107.1,2 It can

be seen from Fig. 7 that In around 105< S< 107 is about

10�2. Therefore, assuming nr/a� 5%, it is expected that the

magnetic island of w/a� 2% exists.

Next, using the index Iw, we compare the behavior of the

magnetic island formation in the resistive interchange instabil-

ity with that in the tearing instability. As shown in Fig. 2(a),

for the first eigenfunction of the resistive interchange instabil-

ity, the value w on the rational surface is much smaller than

the peak w value. On the contrary, as shown in Fig. 8(a), for

the tearing instability, the value w on the rational surface is

about half of the peak w value. This result coincides with the

conventional understanding that the tearing instability forms

the magnetic island larger than the resistive interchange insta-

bility. Figure 10 shows Iw as a function of the S number for the

resistive interchange instability and the tearing instability. It

can be seen that Iw,T is almost the same as Iw,I2. The index Iw,I

decreases with increase in the b value. The decrease of Iw,I1 in

the S number is large for high b plasmas as long as the ideal

interchange instability is dominant.

FIG. 7. An index of the magnetic island formation, In, of the eigenfunctions

with (a) the largest and (b) the second largest growth rates for various b plas-

mas as a function of S.

FIG. 8. Radial profiles of the eigen-

functions (a) w, (b) j, and (c) nr of the

typical tearing instability.

052502-6 Ueda et al. Phys. Plasmas 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

V. CONCLUSION

Focusing attention on the magnetic island formation, we

numerically investigate the characteristics of the resistive

interchange MHD instabilities, which would limit the high boperational regime in the helical type fusion reactors. Up to

now, there are many early works on the perturbed plasma

current and the consequent island formation due to the resis-

tive interchange instabilities. However, typically, the charac-

teristics of the magnetic island formation were studied by the

non-linear simulations of the MHD models and the analyzed

plasma parameter ranges are limited on the b and the mag-

netic Reynolds number (S number).

In this paper, we introduce a new index In, the ratio of

the magnetic fluctuation level to the radial displacement, to

characterize the magnetic island formation. Using the index

In, we systematically analyze the characteristics of the mag-

netic island formation due to the resistive interchange insta-

bilities during the linear growth phase. The analysis is

performed for the m/n¼ 1/1 interchange instability in the

LHD-like configuration, which is observed in the peripheral

region of the LHD discharges in a wide range of the b value

and the S number. Here, m and n are the poloidal and the to-

roidal mode numbers of the MHD activities, respectively.

For the linear analysis, the reduced MHD equations are

solved as the eigenvalue problem. This approach has an

advantage that some eigenfunctions with different eigen-

value (growth rate) can be analyzed, that is, the eigenfunc-

tions with the second and the third largest growth rates can

be analyzed in addition to that with the largest growth rate.

The index In is proportional to the ratio of the square of the

normalized magnetic island width to the maximum absolute

value of the normalized radial displacement. This index ena-

bles us to compare the formed magnetic island width among

the instabilities at the same amplitude of the radial

displacement.

Using the indexes In for the resistive interchange insta-

bility with the largest and the second largest growth rate, we

obtain the following results. Here, it should be noted that the

second largest growth rate is the same order as the largest

growth rate. (i) The formed magnetic island width due to the

second fastest growing instability is larger than that due to

the fastest growing instability when they have the same am-

plitude of the radial displacement as each other. (ii) The

index In for the fastest growing interchange instability is pro-

portional to S�1=3 when the ideal interchange instability is

negligible, and is proportional to S�1 when the ideal instabil-

ity is definite. On the contrary, In for the second fastest grow-

ing interchange instability is proportional to S�1=3 in a wide

range of the b value. (iii) The formed magnetic island width

due to a typical tearing instability is smaller than that due to

the second fastest growing instability of the resistive inter-

change instability when these instabilities have the same am-

plitude of the radial displacement as each other. Also, it is

shown that the magnetic island width due to the tearing

instability is almost the same as that due to the fastest grow-

ing resistive interchange instability.

On the contrary, another index of the magnetic island

formation Iw, the ratio of the poloidal magnetic flux at the

resonant rational surface to that at the maximum absolute

value, shows that the tearing instability has the magnetic

island width much larger than the fastest growing inter-

change instability. The magnetic island width due to the tear-

ing instability is almost the same as that due to the second

fastest growing interchange instability, which supports the

conventional understandings on the magnetic island forma-

tion for the tearing instability and the interchange instability.

ACKNOWLEDGMENTS

This work was performed under the auspices of NIFS

Collaboration Research program (NIFS12KNWP002 and

NIFS13KLPP031). This work was also supported by the

Ministry of Education, Culture, Sports, Science and

Technology, Grant-in-Aid for Scientific Research (B),

25289342.

APPENDIX: CHARACTERISTICS OF IDEALINTERCHANGE INSTABILITY

The ideal interchange instability of the cylindrical

plasma appears when the b value exceeds the Suydam limit.

For such an ideal unstable plasma, it is difficult to analyze

FIG. 9. An index of the magnetic island formation, In, of the interchange

instabilities and the tearing instability as a function of S.

FIG. 10. An index of the magnetic island formation, Iw, of the interchange

instabilities and the tearing instability as a function of S.

052502-7 Ueda et al. Phys. Plasmas 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.

only the resistive interchange instability because the ideal

interchange instability appears simultaneously. This is why

we must also investigate the characteristics of the ideal inter-

change instability. The ideal MHD instabilities can be calcu-

lated by substituting �g to 0 in Eqs. (5)–(7). Figure 11 shows

that the radial perturbation profile of the first and the second

eigenfunction of w. The configuration parameters are the

exactly same as that in Sec. III. However, the b value is

given as bc¼ 4.0% which is much larger than Suydam limit,

bc¼ 2.3%. The growth rates of these eigenfunctions are

1.06� 10�2, 7.05� 10�4, respectively. It should be noted

that the growth rate of the second eigenfunction is much dif-

ferent from that of the first eigenfunction. It can be seen

from Fig. 11(a) that the radial perturbation profile of w in the

first eigenfunction is the odd structure with respect to the

rational surface. On the other hand, the radial perturbation

profile of the second eigenfunction is the even and singular

structure with respect to the rational surface. Here, it should

be noticed that the w is substantially zero at the position of

the rational surface. Therefore, the X-point or the magnetic

islands do not occur on the rational surface. This supports

the fact that the magnetic reconnection does not occur in the

ideal plasma. Figure 12 shows that the radial perturbation

profile of the first and the second eigenfunctions of j around

the rational surface. We can see that the first eigenfunction

of j is almost the odd structure and the directions of the per-

turbed currents are opposite in the vicinity of the rational

surface. On the other hand, the radial perturbation profile of

the second eigenfunction of j is the even structure with

respect to the rational surface and the singular large current

appears on the rational surface. Figure 13 shows the radial

perturbation profile of the first and the second eigenfunctions

of nr . It can be seen that each eigenfunction has a singular

structure on the rational surface. The first eigenfunction has

an extremal point and the even structure with respect to the

rational surface. On the other hand, the second eigenfunction

has two extremal points and the odd structure with respect to

the rational surface.

1K. Y. Watanabe, S. Masamune, Y. Takemura, H. Funaba, S. Sakakibara,

F. Watanabe, K. Tanaka, S. Ohdachi, K. Toi, Y. Narushima, and The LHD

Experimental Group, Phys. Plasmas 18, 056119 (2011).2S. Sakakibara, K. Y. Watanabe, Y. Suzuki, Y. Narushima, S. Ohdachi, N.

Nakajima, F. Watanabe, L. Garcia, A. Weller, K. Toi, I. Yamada, K.

Tanaka, T. Tokuzawa, K. Ida, H. Yamada, A. Komori, O. Motojima, and

The LHD Experimental Group, Plasma Phys. Control. Fusion 50, 124014

(2008).3A. Isayama, S. Inagaki, K. Y. Watanabe, Y. Narushima, S. Sakakibara, H.

Funaba, K. Ida, Y. Nagayama, H. Yamada, K. Kawahata, A. Komori, O.

Motojima, and The LHD Experimental Group, Plasma Phys. Control.

Fusion 48, L45 (2006).4F. Watanabe, K. Toi, S. Ohdachi, S. Takagi, K. Y. Watanabe, S. Morita,

K. Narihara, K. Tanaka, K. Yamazaki, and The LHD Experimental Group,

Plasma Phys. Control. Fusion 48, A201 (2006).5H. P. Furth, J. Killeen, and M. N. Rosenbluth, Phys. Fluids 6, 459 (1963).6K. Ichiguchi, Y. Nakamura, M. Wakatani, N. Yanagi, and S. Morimoto,

Nucl. Fusion 29, 2093 (1989).7K. Ichiguchi, N. Nakajima, M. Wakatani, B. A. Carreras, and V. E. Lynch,

Nucl. Fusion 43, 1101 (2003).8K. Ichiguchi and B. A. Carreras J. Plasma Fusion Res. Series 6, 589 (2004).9H. Miura, N. Nakajima, T. Hayashi, and M. Okamoto, AIP Conf. Proc.

871, 157 (2006).10J. Varela, K. Y. Watanabe, and S. Ohdachi, Phys. Plasmas 19, 082512

(2012).11B. A. Carreras, V. E. Lynch, H. Zushi, K. Ichiguchi, and M. Wakatani,

Phys. Plasmas 5, 3700 (1998).12K. Y. Watanabe, S. Sakakibara, Y. Narushima, H. Funaba, K. Narihara, K.

Tanaka, T. Yamaguchi, K. Toi, S. Ohdachi, O. Kaneko, H. Yamada, Y.

Suzuki, W. A. Cooper, S. Murakami, N. Nakajima, I. Yamada, K.

Kawahata, T. Tokuzawa, A. Komori, and The LHD Experimental Group,

Nucl. Fusion 45, 1247 (2005).13K. Y. Watanabe, S. Sakakibara, H. Funaba, S. Ohdachi, K. Tanaka, I.

Yamada, L. Garcia, Y. Narushima, C. Michel, H. Yamada, Y. Suzuki, N.

Nakajima, K. Narihara, T. Tokuzawa, K. Kawahata, and The LHD

Experimental Group, in Proceedings of the Joint Conference of 17thInternational Toki Conference and 16th International Stella./HeliotronWS, Toki, 2007, 15–19 October 2007, pp. I–13.

14H. R. Strauss, Plasma Phys. 22, 733 (1980).

FIG. 11. Radial profiles of an eigenfunction w of an ideal interchange insta-

bility with the largest and the second largest growth rates.

FIG. 12. Radial profiles of an eigenfunction j of an ideal interchange insta-

bility with the largest and the second largest growth rates.

FIG. 13. Radial profiles of an eigenfunction nr of an ideal interchange insta-

bility with the largest and the second largest growth rates.

052502-8 Ueda et al. Phys. Plasmas 21, 052502 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.